With the recent total solar eclipse, it revived lots of thought of Earth’s ecliptic plane. In terms of forcing, having the Moon temporarily in the ecliptic plane and also blocking the sun is not only a rare and (to some people) an exciting event, it’s also an extreme regime wrt to the Earth as the combined reinforcement is maximized.

In fact this is not just any tidal forcing — rather it’s in the class of tidal forcing that has been overlooked over time in preference to the conventional diurnal tides. As many of those that tracked the eclipse as it traced a path from Texas to Nova Scotia, they may have noted that the moon covers lots of ground in a day. But that’s mainly because of the earth’s rotation. To remove that rotation and isolate the mean orbital path is tricky.  And that’s the time-span duration where long-period tidal effects and inertial motion can build up and show extremes in sea-level change. Consider the 4.53 year extreme tidal cycle observed at the Bay of Fundy (see Desplanque et al) located in Nova Scotia. This is predicted if the long-period lunar perigee anomaly (27.554 days and the 8.85 absidal precessional return cycle) amplifies the long period lunar ecliptic nodal cycle, as every 9.3 years the lunar path intersects the ecliptic plane, one ascending and the other descending as the moon’s gravitational pull directly aligns with the sun’s.  The predicted frequencies are 1/8.85 ± 2/18.6 = 1/4.53 & 1/182, the latter identified by Keeling in 2000.  The other oft-mentioned tidal extreme is at 18.6 years, which is identified as the other long period extreme at the Bay of Fundy by Desplanque, and that was also identified by NASA as an extreme nuisance tide via a press release and a spate of “Moon wobble” news articles 3 years ago.

What I find troubling is that I can’t find a scholarly citation where the 4.53 year extreme tidal cycle is explained in this way. It’s only reported as an empirical observation by Desplanque in several articles studying the Bay of Fundy tides. 

What is intriguing about this 4.53 year cycle is in how it perhaps gets conflated with the half-8.85 year cycle, which being 4.425 years is close to 4.53. There are many references to extreme tidal cycles being dominated either by (1) the 18.6 year nodal declination cycle or (2) the 4.4 year half-perigean cycle. Articles typically assign one as a moon declination effect and the other is a moon distance effect. The half-perigee is explained as a precession of the apsides, in that an additional apogee occurs when the moon is on the opposite side of the planet from where the strongest direct apogee is. So the moon “sees” through the earth, or if tidal forces are partially tangential and thus tractive to the surface, an apogee would alternate on each horizon. This also means that the 27.5545 day anomalistic period has a virtual harmonic at 1/2 that cycle. One such recent reference is an article on nuisance flooding (NF) 

The role of long-period tides in modulating NF frequency​

In addition to secular changes in NF frequencies due to tidal changes, we also detect a ~4.5-year cycle in the time series of additional/reduced NF days (fig. S3), which is close to the half perigee cycle that modulates tides at a period of 4.4 years. The perigean (4.4 years) and nodal (18.6 years) modulations of tides affect high water levels (10). We quantify the impacts of these cycles on NF events by removing them from the tidal prediction and recalculating the number of NF days (assuming that these cycles would not exist); this is only done for the dataset with the observed tides, not the one with historic tides, as we assume that changes in the amplitudes of the 4.4- and 18.6-year cycles were negligible. The oscillations in NF days due to the low-frequency tidal modulations are evident, and their influence increases over time (Fig. 4D). The 4.4-year cycle adds up to 20 NF days across all locations (Fig. 4D) when it peaks under present-day sea level, whereas the 18.6-year cycle causes an additional 30 NF days during its peak compared to average conditions (Fig. 4D)

As with the Bay of Fundy observation, the consensus model and conventional wisdom points to 4.4 years, but the measurements are closer to 4.5 years. The behavioral difference between the two is that 4.53 years depends on the ecliptic plane & perigee whereas 4.42 years is the declination from the equatorial plane & perigee.

( As an aside, what I find odd about this is the disparity in rough precision and detail for this attribution in contrast to the incredible detail needed for total solar eclipse calculations. Consider the grade school science teacher that promised his class in 1978 that they would meet up again in 2024 to watch the eclipse near their school location in upstate New York — and they did just that. Incredible precision needed for that.)

Another piece of evidence is the plotting of data from the NASA JPL Horizons on-Line ephemeris system called Horizons. I didn’t create this plot (courtesy of an Ian Wilson blog post) but I initially didn’t doubt that it is correct, since the JPL tool is widely used for satellite orbit modeling.

So I was confident that I could recreate the chart using Horizons, but alas couldn’t duplicate the 4.53 year modulation cycle that Ian Wilson had managed to find. Instead, after multiplying the delta R lunar distance by the declination output, the best I could match to is the aforementioned 4.42 cycle (along with the 18.6 year cycle). So either Wilson made a mistake (unlikely) or is including a strong solar ecliptic factor wrt to the sun as I am arguing for above.

It may be that some sort of coordinate transformation can coax the NASA JPL Horizons output to align with an ecliptic orientation rather than the geocentric equatorial orientation it does now. I think it requires applying a rotation matrix using the obliquity ϵ (about 5º), where z is the rotation axis and x-y are in the equatorial plane, while the primed values are relative to the ecliptic plane:

x′=x
y′=y⋅cos(ϵ)−z⋅sin(ϵ)
z′=y⋅sin(ϵ)+z⋅cos(ϵ)

Haven’t tried that yet but it appears that it might be able to dilate the time axis by as much as sin(5º), but barring the outcome, the most efficient approach is to make an alternate model of the ecliptic-centered behavior.

First we create a sinusoidal multiplier

ModulatedDraconic = (1 + k sin(2π/18.6 ⋅ t)) * sin(ω_D ⋅ t)

where ω_D is the draconic radial frequency. This when expanded will create 3 sinusoidal terms — the draconic, and 2 satellite subbands. The slower -subband is just the tropical/sidereal lunar period, which makes sense because it represents the value if there were no retrograde speed-up due to the nodal precession. But the obscure part is the faster +subband, which is at 27.1036 days. It has to exist to compensate for the tropical, otherwise averaging the ModulatedDraconic mixed sinusoid wouldn’t generate the mean 27.2122 day period.

Like the 4.53 year period described earlier, this lunar long-period cycle 27.1036 has no name as far as I can tell^[1], yet it indirectly also pops up everywhere in extreme tidal analysis, i.e. 1/27.1036 – 1/27.5545 = 1/(365.242*4.534). Neither of these values appear in R.D.Ray’s table of long period tides for characterizing LOD (below)

Highlighted is one that is close but with reversed Doodson argument where p=1 and N’=2 would produce the 4.53 year period. Yet neither is the 4.42 listed (8.85 is directly above the highlighted) which would be Doodson argument p=2. So this likely reflects the fact that these are not an important feature for LOD as opposed to tides. This may have some implications for extending extreme tides to modeling oceanic climate indices, as I have perhaps put too much of a focus on LOD calibration in modeling ENSO and AMO, which is considered next.

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The same ecliptic considerations apply to the QBO, where I published that the frequent crossing of the lunar  orbit with the ecliptic plane, at 27.212/2 day intervals constructively interferes with the semi-annual cycle, thus maximizing gravitational forcing  (moon+sun) at the frequency observed of QBO.  Thus 365.242/27.2122 mod 1 = 0.422 is the 2.368 quasi-biennial period.   And again nowhere will you find a citation to this identity — unless to the linked pub.

The connection is that I also recently used the 27.1036 day value on the QBO to explain the anomalies observed. When used there with the main draconic cycle and a slight compensating tropical cycle it can maintain the 2.368 year QBO cycle while fluctuating about the average period.

But now what’s in store is that it also appears to be able to model the Bay of Fundy’s big neighbor — the AMO. The 70 year multidecadal period naturally falls out, at it is related to the 27.1036 day value as well. In this case it is 365.242/27.106 + 2*365.242/27.5545 mod 1 = 0.986, which will create a long-term modulation when lag integrated against the annual cycle, so that 1/(1-0.98) ~ 70. In the past I had been using the Mf and Mm (2/27.3216 + 1/27.5545) combination to create a 120 year period, which could only work if LTE harmonics were applied.

So more to come — thanks to thinking about the solar eclipsing moon tracing it’s path to the Bay of Fundy ^[2] for the recent insight.

Footnotes

[1] I’d like to call this new period either the Laciport or Cinocard month which is the backward spelling of Tropical or Draconic.

[2] Desplanque, Con, and David J. Mossman. “Tides and their seminal impact on the geology, geography, history, and socio-economics of the Bay of Fundy, eastern Canada.” Atlantic Geology 40.1 (2004): 1-130